Estimation of Median Incomes of the American States: Bayesian Estimation of Means of Subpopulations

Abstract

The Fay–Herriot model (J Am Stat Assoc 74:269–277, 1979) is an immensely popular model in small area estimation of means of some characteristics for many related subpopulations. This model regresses the subpopulation mean on a set of auxiliary variables to borrow strength from other subpopulations. The Fay–Herriot model uses a fixed regression coefficient vector but only a random intercept term to account for variability that cannot be explained by the non-random mean function. In some applications, the non-random regression coefficient often does not adequately account for the variability of the subpopulation means. To accommodate extra variability, we consider an extension of the random intercept model by treating some of the regression coefficients also as random. This model is referred to as the random regression coefficient model. For the flexible random regression coefficient model, we allow the intercept and some of the regression coefficients to randomly vary across states with a suitable normal distribution. We use a suitable noninformative prior for all the model parameters to conduct our Bayesian analysis. We establish propriety of the resulting posterior density function and generate Monte Carlo samples from this distribution to get point estimates and associated measures of accuracy of these estimates. We also construct relevant credible intervals of the small area means as another measure of uncertainty for the point estimates. The method is illustrated to predict four-person family median incomes for 1989 for the US states.